noxpordfr - Metamath Proof Explorer

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Theorem noxpordfr 28364

Description: Next we establish the foundedness of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.)

**Hypotheses**
Ref	Expression
noxpord.1	⊢ 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))}
noxpord.2	⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (No × No) ∧ 𝑦 ∈ (No × No) ∧ (((1^st ‘𝑥)𝑅(1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥)𝑅(2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}

**Assertion**
Ref	Expression
noxpordfr	⊢ 𝑆 Fr (No × No)

Distinct variable groups: 𝑥,𝑅,𝑦 𝑎,𝑏 Allowed substitution hints: 𝑅(𝑎, 𝑏) 𝑆(𝑥, 𝑦, 𝑎, 𝑏)

**Proof of Theorem noxpordfr**
Step	Hyp	Ref	Expression
1		noxpord.2	. . 3 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (No × No) ∧ 𝑦 ∈ (No × No) ∧ (((1^st ‘𝑥)𝑅(1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥)𝑅(2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}
2		noxpord.1	. . . . 5 ⊢ 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ ((L‘𝑏) ∪ (R‘𝑏))}
3	2	lrrecfr 28356	. . . 4 ⊢ 𝑅 Fr No
4	3	a1i 11	. . 3 ⊢ (⊤ → 𝑅 Fr No)
5	1, 4, 4	frxp2 8185	. 2 ⊢ (⊤ → 𝑆 Fr (No × No))
6	5	mptru 1577	1 ⊢ 𝑆 Fr (No × No)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∨ wo 861 ∧ w3a 1103 = wceq 1570 ⊤wtru 1571 ∈ wcel 2147 ≠ wne 2959 ∪ cun 4152 class class class wbr 5106 {copab 5170 Fr wfr 5605 × cxp 5653 ‘cfv 6555 1^st c1st 8020 2^nd c2nd 8021 Nocsur 28024 Lcleft 28238 Rcright 28239

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-1st 8022 df-2nd 8023 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-1o 8501 df-2o 8502 df-no 28027 df-lts 28028 df-bday 28029 df-slts 28171 df-cuts 28173 df-made 28240 df-old 28241 df-left 28243 df-right 28244

This theorem is used by: norec2fn 28369 norec2ov 28370

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